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Dr Oksana Shatalov, Fall 2012
7.1: Areas Between Curves
One of interpretations of definite integral
Z b f ( x ) d x, f ( x ) ≥ 0 on [ a, b ] a is the area between the graph of y = f ( x ) and the x -axis on [ a, b ].
y
0
For example, if f ( x ) = cos x and x ∈ [0 ,
π
2
] then x
1
Z b
If f ( x ) ≥ 0 on [ a, b ] then f ( x ) d x a
Z b
If f ( x ) ≤ 0 on [ a, b ] then f ( x ) d x a
The previous example on [0 ,
2 π
3
] :
Our goal: Find the area between two curves.

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Dr Oksana Shatalov, Fall 2012 2
CASE I.
Determine the area between y = f ( x ) and y = g ( x ) on the interval [ a, b ] assuming f ( x ) ≥ g ( x ) on [ a, b ] .
I other words, find the area of the region D defined by y x
0
Solution:
Explanation :
A = A ( D ) =
Z b f ( x ) − g ( x ) d x a

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Dr Oksana Shatalov, Fall 2012 3
CASE II.
Determine the area between x = f ( y ) and x = g ( y ) on the interval [ c, d ] assuming f ( y ) ≥ g ( y ) on [ c, d ] .
I other words, find the area of the region D defined by y x
0
Solution:
A = A ( D ) =
Z d f ( y ) − g ( y ) d y c
The above formulas in the ”word” form:
CASE I A =
Z b a upper function
!
− lower function
!
d x
CASE II A =
Z d c right function
!
− left function
!
d y
Coming back to the previous example: f ( x ) = cos x , where 0 ≤ x ≤ 2 π/ 3 we get:

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Dr Oksana Shatalov, Fall 2012 4
EXAMPLE 1 .
Determine the area of the region enclosed (=bounded by) by y = x 2 and y =
√ x .
y x
0
REMARK 2 .
1. The limits of integration in the above example were determined as the intersection points of the two curves.
2. Sketch of a graph of the region is recommended (it helps to determine which of the functions is upper/right).
3. The area between two curves will always be
EXAMPLE 3 .
Determine the area of the region bounded by y =
1 x and y = − 1 , x = 1 , x = 3 .
y x
0

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Dr Oksana Shatalov, Fall 2012
EXAMPLE 4 .
Determine the area of the region bounded by y = 2 x 2 + 4 and y = 4 x + 10 .
y
5 x
0
EXAMPLE
5
.
Determine the area of the region bounded by y = 2 x 2 + 4 and y = 4 x + 10 , x = − 2 , x = 5 .
y x
0

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Dr Oksana Shatalov, Fall 2012 6
EXAMPLE 6 .
Determine the area of the region enclosed by y = sin x , y = sin 2 x , x = 0 , x = π/ 2 .
y x
0

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Dr Oksana Shatalov, Fall 2012
EXAMPLE 7 .
Determine the area of the region enclosed by x =
1
2 y 2 − 3 , y = x − 1 .
y
7 x
0
EXAMPLE
8
.
Determine the area of the region bounded by the x -axis, the curve y = x 2 tangent line to this curve at the point (1 , 1) .
and y x
0